Basis and dimension

• Sep 25th 2010, 10:37 PM
acevipa
Basis and dimension
Find the basis for and the dimension of $W=span(\mathbf{v_1, v_2, v_3})$, where

$\mathbf{v_1}=\begin{pmatrix}1\\ 2\\ 3\end{pmatrix}, \mathbf{v_2}=\begin{pmatrix}1\\ 1\\ -1\end{pmatrix}, \mathbf{v_3}=\begin{pmatrix}-1\\ 0\\ 5\end{pmatrix}$

I'm not too sure how to start this question off
• Sep 26th 2010, 08:43 AM
HallsofIvy
If the three given vectors are independent, then they are a basis for their own span and the dimension is 3. If they are not, then one of the vectors can be written as a linear combination of the other two and so those two form a basis and the dimension is 2. (It clear that these vectors are not all multiples of one another so the dimension is not 1.)

So start by checking if they are independent or not: do there exist numbers, a, b, and c such that
$av_1+ bv_1+ cv_2= a\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}+ b\begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}+ c\begin{pmatrix}-1 \\ 0 \\ 5\end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0\end{pmatrix}$

That is the same as
$\begin{pmatrix}a \\ 2a \\ 3a\end{pmatrix}+ \begin{pmatrix}b \\ b \\ -b\end{pmatrix}+ \begin{pmatrix}-c \\ 0 \\ 5c\end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$

$\begin{pmatrix}a+ b- c \\ 2a+ b \\ 3a- b+ 5c\end{pmatrix}= \begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}$

and that is the same as the three equations a+ b- c= 0, 2a+ b= 0, and 3a- b+ 5c= 0. Obviously a= b= c= 0 is a solution. If it is the only solution, then the three vectors form a basis and the space is three dimensional.

If there exist another solution, we can solve for one of the vectors in terms of the other two. If we had $av_1+ bv_2+ cv_3= 0$ with, at least, c non-zero, then we can write $cv_3= -av_1- bv_2$ so $v_3= -(a/c)v_1- (b/c)v_2$. Since $v_1$, $v_2$, and $v_3$ span the space every vector in it can be written as a linear combination, $v= pv_1+ qv_2+ rv_3$. With $v_3= (-a/c)v_1- (b/c)v_3$ we can replace $v_3$ and get $v= pv_1+ qv_2+ r(-(a/c)v_1- (b/c)v_2)= (p- ar/c)v_1+ (q- br/c)v_2$. Then $\{v_1, v_2\}$ is a basis and the space has dimension 2.

That's the idea. I'll leave the actual calculation to you.